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Chapter 6: Problem 273

In the following exercises, solve each proportion. $$ \frac{b}{-7}=\frac{-30}{42} $$

Short Answer

Expert verified
b = 5

Step by step solution

01

Set up the cross multiplication

To solve the proportion \(\frac{b}{-7}=\frac{-30}{42}\), use the cross multiplication method. Cross multiplying \(\frac{b}{-7}\) and \(\frac{-30}{42}\) will give us: \(b \cdot 42 = -7 \cdot -30\).
02

Simplify the equation

Simplify the right-hand side of the equation. \(-7 \cdot -30 = 210\) because multiplying two negative numbers yields a positive product. Therefore, the equation becomes: \(42b = 210\).
03

Solve for b

Isolate \(b\) by dividing both sides of the equation by 42. \(b = \frac{210}{42}\).
04

Simplify the fraction

Simplify \(\frac{210}{42}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 42. \(\frac{210 \div 42}{42 \div 42} = 5\). So \(b = 5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cross Multiplication
Cross multiplication is a powerful method used to solve proportions. When you have a proportion like \(\frac{a}{b}=\frac{c}{d}\), cross multiplying helps by setting up an equation that is easier to solve. In this method, you multiply the numerator of one fraction by the denominator of the other fraction and set the results equal to each other. This will look like: \[a \times d = b \times c\].
For our example, \(\frac{b}{-7}=\frac{-30}{42}\), cross multiplication gives us: \[b \times 42 = -7 \times -30\].
This method is excellent because it transforms a potentially complex proportion into a straightforward equation.
Simplifying Equations
Simplifying equations is an essential skill in algebra. After cross multiplying, you'll usually get an equation that needs further simplification. To simplify means to make the equation easier to solve.
For example, from our cross multiplication step, we obtained: \[b \times 42 = -7 \times -30\].
First, we multiply the terms on the right: \[ -7 \times -30 = 210 \] because multiplying two negative numbers gives a positive result. So our equation simplifies to: \[42b = 210\].
To solve for \(b\), divide both sides of the equation by 42: \[b = \frac{210}{42}\].
Simplification helps us to more clearly see what the variable equals.
Greatest Common Divisor
The Greatest Common Divisor (GCD) is useful when simplifying fractions. The GCD of two numbers is the largest number that divides both of them without leaving a remainder.
Let's see how it works with our fraction \( \frac{210}{42} \). We want to simplify this fraction by dividing both the numerator and denominator by their GCD.
First, find the GCD of 210 and 42, which is 42.
  • \( \frac{210}{42} = \frac{210 \div 42}{42 \div 42} \).
This gives us \( \frac{5}{1} = 5 \).
Hence, \( b = 5 \).
Using the GCD to simplify fractions ensures that the fraction is reduced to its simplest form quickly and efficiently.

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